There is no historical evidence that the ancient Greeks ever measured the frequencies of sounds. They developed their theories of intervals using the relative lengths of strings or pipes, which are an equivalent way to relate intervals to geometrical ratios of lengths.
There is no obvious reason why the Greeks could not have invented the siren (see below), and by 300BC they already had reasonably accurate water-driven clocks, but in general they were more interested in philosophical discussion about how things ought to work, rather than scientific experimentation to find out how they actually worked.
The method of sound frequency measurement in Hz used by Helmholtz in "The Sensations of Tone" (first published 1863) was a siren, which is basically a rotating disk with a pattern of holes through which air is blown. It is easy to measure the rotational speed using a clock, and hence know the fundamental frequency of the sound produced.
I remember seeing a demonstration of exactly the same measurement system in high-school-level physics, back in the 1960s. Of course electronic frequency measurement was possible at that time, but (like modern tuning apps) it was a "magic black box" compared with a mechanical system whose working could be understood.
Helmholtz's other experimental method was to use sympathetic resonance to demonstrate the existence of a particular overtone in a sound, but that does not measure the frequency in Hz except by an independent calibration of the resonator itself.
Before Helmholtz, pitch standards (pipes, bells, tuning forks, etc) were simply compared with each other, and not with any absolute measurement. As a result, almost every geographical area in Europe had is own local pitch standard, though travelling professional musicians, and the practical range of human voices, tended to keep them more or less aligned with each other.
A mechanical strobe tuner is not a practical method of tuning musical instruments. The tuning fork was invented in 1711, and before that time the usual pitch standards were bells or reed pipes.
For church music, pipe organs provided a "standard" simply because they were impossible to retune quickly. Ironically the pitch of organ flue pipes (similar to a flute or recorder) is temperature dependent while the pitch of reed pipes is not, but in practice the reed pipes were (and still are) retuned to match the flues to compensate for temperature changes, because that is more practical.